5.3 Efficient right hand side
5.3.1 VEM interpolant
For VEM we have a different approach, using that Vk(E) contains more functions than only
polynomials. Hence, we consider approximating f with some fI ∈Vk(E) and use that for the
computation ofFE.
To definefI, we use that the degrees of freedom (χi(·)) implicitly define a natural interpolation.
The interpolant is defined as the functionfI ∈Vk(E) that satisfies
χi(f) =χi(fI), i= 1,2, . . . , nd.
Where the moment degrees of freedom can be computed by using triangulation and quadra- ture, or by using the divergence theorem and integration over the boundary of E. Using the
CHAPTER 5. DGVEM 5.3. EFFICIENT RIGHT HAND SIDE
interpolantfI as approximation off inFE we get
FE i = Z E φifdx≈ Z E φifIdx.
To see if we can compute this, we observe that fI ∈ Vk(E) and thus we can express it as a
vector with the values of its degrees of freedomfI. Then the integral is equivalent to
Z E φifI = nd X j=1 ME(φ i, φj)fI,i, with ME(φi, φj) = Z E φiφjdx.
Using the standard polynomial basis functions the computation of these integrals is no problem resulting in the mass matrix. For VEM this is, of course, a bit more complicated and we need the approximation from [3]. Using the same steps as with the stiffness matrix they propose
ME(φi, φj)≈Mij =ME(Π0kφi,Π0kφj) +S0E(φi−Π0kφi, φj−Π0kφj).
Where nowSE
0 is a suitable symmetric bilinear stabilization term similar toSE for the stiffness
matrix. Using this computable approximation gives
FE=MfI. (5.11)
Thus, we can use this alternative approach for the computation of the right hand side.
5.3.2
Comparison
With these three approaches (Polynomial projection, Taylor polynomial, VEM interpolation) we have the question which one to use. For ADG we already argued that the Taylor polynomial is probably the least expensive and exact enough for the right hand side computation. The question is thus how does the VEM interpolation compare to both methods.
For this we have to consider the cost. To find the VEM projection fI we need to compute
moments of order up tok−2 and the value at the boundary degrees of freedom. Typically this is cheaper than the computation of all the k moments off for the polynomial projection, as moments are more expensive to compute than the value off. In addition, we have to compute the mass matrix approximation M, which is not a cheap operation, as it requires inverting
IE.
When considering the gains in accuracy, we only know that fI is a function in Vk(E). This
space contains Pk(E), which contains the polynomial projection of f, thereby allowing for a
more accurate approximation. However, we do not know iffIis actually a better approximation,
and whether this accuracy is lost when using the approximated mass matrix.
Comparing VEM interpolation to the Taylor polynomial approach we can be brief. As VEM interpolation requires the computation of both moments off and the approximated mass matrix it will be much more expensive. Therefore, we expect that the Taylor polynomial, assuming sufficiently accurate, is the better approach.
Comparing VEM interpolation to polynomial projection, we expect that the use of the VEM interpolant might be computationally beneficial for problems where we already computed the mass matrix. For such problems, we do not incur the extra penalty of computing the mass matrix, while we reduce the number of moments off that we need to compute.
5.4. ERROR COMPUTATION CHAPTER 5. DGVEM
5.4
Error computation
Just as with ADG, we also need to look at the error computation. For VEM this is even a bigger problem, as not only do we have the product between a virtual solutionuh and the unknown
function u, we also have the problem of integrating the product u2
h. We investigated several
options.
5.4.1
Polynomial projection
The first option is to use a polynomial projection, in this case of the VEM solution. By computing Π0
kuh we get a polynomial approximation of the VEM solution, which we can use
for the error computation.
Here we can choose two directions. The simplest is to triangulate the element and then perform a quadrature to compute Z E u−Π0 kuh 2 dx. (5.12)
An approach that is very straightforward in implementation, but not very cheap to com- pute.
Alternatively, we can also project uto Pk(E) using Pk. Using this approach, the integration
becomes almost trivial as bothPkuand Π0kucan be expressed as monomial coefficients. Letud
be the vector with the differences in the coefficients for the scaled monomials, and we have
Z
E
Pku−Π0ku
2
dx=uTdIEud. (5.13)
This can be faster (depending on the speed of computing the moments ofu), but throws away even more detail of the solution. Projectinguto a higher polynomial space is of course possible, but increases both the cost of the projection and the cost of computing the largerIE.
In the extreme case, we could forgo all attempts at creating a good error approximation and use as projectionPkuthe local Taylor polynomial ofu. Though, at that point one can certainly
argue that we are stacking so many approximations, that we only get a rough idea of the error size instead of an actual approximation.
5.4.2
VEM interpolation
A better attempt may be found using the VEM interpolantuI, constructed using the same pro-
cess as the VEM interpolationfI off. Using this interpolant, we can computeuI−uh∈Vk(E)
exactly. Moreover, letud be the vector with the degrees of freedom of this difference. Then we
can compute theL2-norm of this difference using the approximate mass matrixM:
Z
E
(u−uh)2dx≈uTdMud.
Using this approach, we are again introducing approximations by first interpolating u to get
uI and then using the approximate mass matrix. However, unlike the polynomial projection
approaches, we are not projecting the solutionuhfromVk(E) to a lower dimensional space like
(Pk(E)). This prevents the loss of accuracy in projecting the solution to Pk(E). How much
CHAPTER 5. DGVEM 5.4. ERROR COMPUTATION
5.4.3
Solution reconstruction
For the third option, we go against the idea of VEM not to compute the basis functions. Using the definition of Vk(E) and the computed degrees of freedom we can reconstruct the
actual solution uh. This solution can then be used to compute the difference u−uh and
compute the error. As the basis functions of VEM are not designed to be computed, this will be computationally expensive and thus not useful for actual large scale problems.
Looking at the requirements for Vk(E) in (5.1.3), we see that our reconstructed solution ¯uh
should satisfy:
• ∆¯uh=pfor somep∈ Pk(E) insideE,
• u¯h|∂E ∈ Bk(∂E), fixing the value on the boundary of E,
• the moments of order k−1 and k for ¯uh and Π∇u¯h should coincide: REu¯hmdx =
R
E Π∇u¯h
mdx form∈ M?
k−1(E)∪ M?k(E).
Moreover, as ¯uh is the reconstruction ofuh, we also expect that the degrees of freedom match,
thusχi(¯uh) =χi(uh).
As we do not know the actual polynomialp in the first constraint, we will need to solve the Poisson equation several times. Hence, we create a mesh ofE and our reconstruction ¯uh will
be an approximation touh from a suitable finite element space.
For the actual computation, we start with the second constraint and constructuh,∂ ∈ Bk(∂E)
using the boundary degrees of freedom. This function is then used as boundary condition and solve ( ∆u0 h= 0 inE, u0 h=uh,∂ on∂E. (5.14) The solution u0
h obviously satisfies the first two constraints that we placed on ¯uh and has
matching boundary degrees of freedom. However, the last constraint is not satisfied, nor is the equality of the moment degrees of freedom. To satisfy both we need that ∆u0
h∈ Pk(E). We do
not know the coefficients of this polynomial, but we can solve
( ∆ui h=mi inE, ui h= 0 on∂E, (5.15) where themi are the usual scaled monomials with i= 1,2, . . . , nP,k. Using these solutions we
define our reconstruction of the solution as ¯ uh=u0h+ nP,k X i=1 αiuih,
where theαi are constants that still have to be determined.
To find the constantsαi we will reuse the definition of Π0k foruh(5.6):
Z E (Π0kuh)mdx= Z E uhmdx, ∀m∈ Mk(E).
The values on the left hand side are computable using the actual degrees of freedom on uh
and the already developed machinery of Π0
5.5. SUMMARY CHAPTER 5. DGVEM approximation ¯uh to get Z E (Π0 kuh)mdx= Z E u0 hmdx+ nP,k X i=1 αi Z E ui hmdx.
The actual values on the right hand side can be computed using standard quadrature on the mesh of E used for computing ¯uh. Plugging in all the monomialsmi ∈ Mk(E) as m, we get
nP,k equations, which is sufficient to find thenP,k constantsαi, thereby fully determining our
reconstruction ¯uh.
After all this work we can finally use the result to compute the errorku−uhk2E ≈ ku−u¯hk2E.
Again, we compute this integral using the mesh forEused in the construction of ¯uhalong with
standard quadrature rules.
5.4.4
Comparison
From all these options we see that the error computations for VEM can only be estimates. We can get arbitrarily close when reconstructing the solution, but at the hefty price of solving
nP,k+ 1 Poisson equations per element. The polynomial and VEM interpolant are faster, but
only give a rough approximation of the error.
Having to solve that many PDE’s per element reduces the practical use of the solution re- construction method. Therefore we do not expect it to be useful for any larger scale error computations. For that we can still use VEM interpolation and polynomial projection, though these methods will introduce some inaccuracy.
Comparison of using the VEM interpolant or the polynomial projection approach is almost pure speculation. The projection of uh to the polynomials for computing its error, also introduces
an error. Though using the VEM interpolant can be better, but introduces errors due to the use of an approximate mass matrix.
5.5
Summary
In this chapter we looked at DGVEM. This method uses the non-polynomial, but conforming basis functions of VEM, combined with the discontinuous Galerking approach of SIPG. By using these VEM virtual basis functions, it also inherits the complications that stem from not computing them. The construction of the element matrix and element vector require two projections of the basis functions, Π∇φ and Π0
kφ, that need to be computed. Moreover, the
non-polynomial functions f and uin the element vector and error computations, respectively, cause more difficulty than with ADG.
For the element vector we can not use the triangulation and quadrature approach, but the polynomial projection and Taylor polynomial options remain possible, with the addition of the VEM interpolant. Just as with ADG, we expect that the Taylor polynomial to be the computationally cheap, but sufficiently accurate option.
For the error computation, we, unfortunately, have less appealing choices. We can project the solution back to polynomials, loosing information and accuracy of the solution. Alternatively, we can interpolate the solutionuto VEM space, but have to use an expensive and only approximate
CHAPTER 5. DGVEM 5.5. SUMMARY
mass matrix. The third alternative of reconstruction ofuhis can be very accurate but hideously
expensive, as it requires solving many Poisson problems on each of the elements in the mesh. This need to solve more Poisson problems on smaller meshes to compute the error, defeats the purpose of using DGVEM to solve our original Poisson problem on a large mesh. Therefore, it is unusable for practical problems, but could be useful for more theoretical purposes.
Chapter 6
Numerical results
With the theoretical side of ADG and DGVEM covered, we can now turn to the numerical results. We start with standard convergence results to the verify the correct behaviour of the method. Certain that our methods work as intended, we then continue by looking at the different element vector and error computation possibilities for both ADG and DGVEM. We finish with the most important part of the numerical results, a comparison between ADG and DGVEM.
For these numerical discussions we will need to fix the accuracy order k for the methods. Therefore, we will write ADGkand DGVEMkto denote the ADG and DGVEM methods using a specific value of k, e.g. ADG1 is ADG withk= 1. Furthermore, unless stated otherwise we will assumeγ= 20 as stabilization parameter andκ= 1.
6.1
Verification
To verify that both ADG and DGVEM work as expected, we test them using two manufactured solutions. For both methods we solve our generalized Poisson model problem (2.1), as domain Ω we use the unit square, with Dirichlet boundary conditions around the full boundary (i.e. ΓD=∂Ω). On this domain we use a mesh of regular triangles, see Figure 6.1.
The first verification is done using the manufactured solution
u(x1, x2) = sin(2πx1) sin(2πx2), (6.1) with as consequence ( f(x1, x2) = 8π2u(x1, x2) in Ω, gD= 0 on∂Ω. (6.2) We computed the solution using the appropriatef for four methods. As baseline we use SIPG in physical coordinate space from Chapter 2. For ADG and SIPG, we use the polynomial projection (4.23), with (4.18) for required moments, for the element vector and quadrature for the error computation. For DGVEM, we use again the polynomial projection (5.9), with (4.22) and (4.18), for the element vector, while the error is computed by projecting the solution to
CHAPTER 6. NUMERICAL RESULTS 6.1. VERIFICATION
(a) Smallest mesh (b) First refinement
Figure 6.1: First two meshes used for the first convergence test.
10−1.5 10−1 10−0.5 10−3 10−2 10−1 h k u − uh k SIPG ADG DGVEM (a)k= 1 10−1.5 10−1 10−0.5 10−5 10−4 10−3 10−2 h k u − uh k SIPG ADG DGVEM (b)k= 2
Figure 6.2: Convergence for the first testcase usingγ = 20. Note that for k = 1 the lines of ADG and DGVEM overlap, while fork= 2 the lines of SIPG and ADG overlap.
6.2. ADG CHAPTER 6. NUMERICAL RESULTS
Table 6.1: Error norm ku−uhk for the second test case, using γ = 20 and the mesh of Fig-
ure 6.1b. The column marked withDis with Dirichlet boundary conditions on the whole bound- ary, while theDN uses Neumann boundary conditions on the boundary part with x2= 0,1.
Method Order L2-Error (D) L2-Error (DN)
SIPG 1 4.5×10− 15 3.3×10−15 2 9.5×10−15 1.3×10−14 ADG 1 4.8×10− 15 1.4 ×10−14 2 6.4×10−15 2.3 ×10−14 DGVEM 1 1.8×10− 14 5.1×10−14 2 2.6×10−14 5.6×10−14
polynomials and using quadrature (5.12). We can see in Figure 6.2 that all three methods converge at the expected ratehk+1.
The second verification is done to test with discontinuousκ, specifically, one of the form
κ= (1 2 forx1≤ 1 2, 10 forx1> 12.
Additionally, we choose as solution,
u= ( 2x1κ1κ+κ2 2 forx1≤ 12, 1−2x1κκ1 1+κ2 forx1> 1 2,
which gives as problem
n
f = 0 in ΩgD= u|∂Ω.
Note that this solution is piecewise linear. Both ADG and DGVEM can theoretically exactly reproduce this solution. In Table 6.1, we see that the error is, as expected, near machine precision.
6.2
ADG
Now that we know that both methods work as intended, we can compare the alternatives for the element vector. We start with ADG, and the alternatives that were introduced in Chapter 4.
To tests these, we need one or more polygonal meshes. We choose to create meshes by using a proto-tile to tile the plane and cut a square section out of it. As result several tiles at the boundary are cut off, creating possibly elements that are small or that have large aspect ratio. Creating more refined meshes is easy, we can simply tile our domain with a smaller proto-tile. Specifically, for our test cases we create smaller meshes by scaling each of the proto-tiles’ edges by a factor 12.
The resulting meshes that are generated (at the largest level), are shown in the first column of Figure 6.3. As can be seen in this figure, we use five proto-tiles: triangle, hexagon, arrow, brick and boat. The triangle, as it is traditionally used in finite element methods. The hexagon, arrow and brick shapes because they all have six vertices, which ensures that ADG2 and DGVEM1
CHAPTER 6. NUMERICAL RESULTS 6.2. ADG
both have six degrees of freedom per element. With both convex (hexagon) and non-convex (arrow) elements and additionally one with hanging nodes (brick). Lastly, the boat shape has two reflex angles, creating a possibly ill-conditioned element.
6.2.1
Norm
As first alternatives for ADG, we consider the norm computation. We consider four alternatives for the computation ofku−uhkE:
1. Traditional quadrature on the possibly triangulated element. To increase the accuracy, we require that the quadrature is accurate for polynomials up to order 2k+ 4.
2. The polynomial projection tok-th order polynomials (4.26). The projection Πkuis com-
puted using (4.22) withl=kand (4.18) to compute the moments. In the last computation we used a line quadrature rule accurate for polynomials of order 2k+ 3.
3. A local Taylor polynomial approximation ofuof orderk. Then using (4.27) for computing the actual error.
4. The same Taylor polynomial approach, but with a polynomial of orderk+ 2. Note that this requires the computation of a largerIE.
In addition to these, we reuse the sine test case from (6.2). The element vector is computed